Problem 73 - Physics

Question

Interactive Solution \(11.73\) at illustrates a model for solving this problem. A pressure difference of \(1.8 \times 10^{3} \mathrm{~Pa}\) is needed to drive water \(\left(\eta=1.0 \times 10^{-3} \mathrm{~Pa} \cdot \mathrm{s}\right)\) through a pipe whose radius is \(5.1 \times 10^{-3} \mathrm{~m}\). The volume flow rate of the water is \(2.8 \times 10^{-4} \mathrm{~m}^{3} / \mathrm{s}\). What is the length of the pipe?

Step-by-Step Solution

Answer

The length of the pipe is approximately 0.107 m.

Step 1: Understand the Problem

We need to find the length of the pipe given the pressure difference, the viscosity of the fluid, the radius of the pipe, and the volume flow rate of the water.

Step 2: Use Poiseuille's Law

Poiseuille's Law relates the flow rate (Q), the pressure difference (abla P), the radius (r), the viscosity (eta), and the length (L) of the pipe:\[ Q = \frac{{\pi r^4 abla P}}{{8 \eta L}} \]

Step 3: Rearrange the Formula to Solve for Length

To find the length (L), rearrange the formula:\[ L = \frac{{\pi r^4 abla P}}{{8 \eta Q}} \]

Step 4: Plug in the Given Values

Substitute the given values into the equation:- Pressure difference \( abla P = 1.8 \times 10^{3} \ \mathrm{Pa} \)- Radius \( r = 5.1 \times 10^{-3} \ \mathrm{m} \)- Viscosity \( \eta = 1.0 \times 10^{-3} \ \mathrm{Pa \cdot s} \)- Flow rate \( Q = 2.8 \times 10^{-4} \ \mathrm{m}^{3}/s \)\[ L = \frac{{\pi \cdot (5.1 \times 10^{-3})^4 \cdot 1.8 \times 10^{3}}}{{8 \cdot 1.0 \times 10^{-3} \cdot 2.8 \times 10^{-4}}} \]

Step 5: Calculate the Length

Calculate the value:\[ L = \frac{{\pi \cdot 6.7321 \times 10^{-14} \cdot 1.8 \times 10^{3}}}{{2.24 \times 10^{-6}}} \]\[ L \approx 0.107 \ \mathrm{m} \]

Step 6: Conclusion

The length of the pipe is approximately \(0.107 \ \mathrm{m}\).

Key Concepts

Poiseuille's Law

In fluid dynamics, Poiseuille's Law is a key principle that describes how viscous fluid flows through a cylindrical pipe. The law provides a relationship between various parameters that affect the flow rate of the fluid. It is expressed mathematically by the equation:

• \[ Q = \frac{{\pi r^4 \Delta P}}{{8 \eta L}} \]

Where:

• \( Q \) is the volumetric flow rate, or how much fluid passes through the pipe per unit of time.
• \( \Delta P \) represents the pressure difference between the ends of the pipe.
• \( r \) is the radius of the pipe.
• \( \eta \) stands for the fluid's viscosity.
• \( L \) indicates the length of the pipe.

Poiseuille's Law is particularly relevant in circumstances where the fluid flow is laminar, meaning it flows in parallel layers without any disruption between them. Understanding this law provides insight into the control and efficiency of fluid delivery in systems such as pipelines, vascular networks in biological organisms, or even simple household plumbing.

Viscosity

Viscosity is a measure of a fluid's resistance to deformation or flow. In simpler terms, it describes how "thick" or "sticky" a fluid is. For example, honey has a higher viscosity compared to water. In the context of Poiseuille's Law, viscosity is a crucial factor because it influences how easily a fluid can move through a pipe.

• Higher viscosity means more resistance, which slows down the flow rate.
• Conversely, lower viscosity fluids flow more freely.

When we look at the formula from Poiseuille's Law, the viscosity \((\eta)\) is in the denominator, showing that as viscosity increases, the flow rate \((Q)\) decreases, assuming all other factors are constant. This relationship helps in designing systems where fluid flow needs to be controlled or predicted, such as medical devices or industrial applications where fluid consistency cannot vary significantly.

Pressure Difference

Pressure difference, denoted as \( \Delta P \), is a fundamental concept in fluid dynamics, influencing the movement of fluids through Pipelines. It refers to the difference in pressure between two points in a pipe.

• A greater pressure difference typically accelerates fluid flow, propelling it from the high-pressure region to the low-pressure region.
• On the contrary, a smaller pressure difference results in slower flow rates.

In Poiseuille's Law, pressure difference is directly proportional to the flow rate \((Q)\), assuming other variables such as viscosity \((\eta)\), pipe length \((L)\), and pipe radius \((r)\) remain constant. This means that when the pressure difference increases, the flow rate tends to increase as well. Engineers and scientists use this principle to manipulate and predict fluid movement in a range of applications, from natural systems to complex industrial processes.

Flow Rate

Flow rate in fluid dynamics is the volume of fluid that moves through a specified area in a given time frame. It is often expressed in cubic meters per second \((\mathrm{m}^3/\mathrm{s})\).

• Flow rate provides valuable information about the efficiency and effectiveness of fluid systems, such as water supply networks or fuel pipelines.
• It helps in determining the optimal design and operation conditions for systems requiring precise fluid management.

Poiseuille's Law illustrates how the flow rate \((Q)\) is a function of several factors: pipe radius, fluid viscosity, pressure difference across the pipe's length, and the pipe's length itself. By understanding how these variables influence the flow rate, engineers can adjust these parameters to meet specific requirements, whether to increase transportation efficiency or develop consistent delivery mechanisms for sensitive fluids.